Fractals: 3D

Into
the Third Dimension

The
Mandelbrot Set can be thought of
as the
result of an
"unstable" function with two parameters, and two zones ... a zone where
the
function obviously goes to infinity,
and a zone where it doesn't. In between the two, there's a
chaotic
region where these two outcomes mix together, and where the result of
the function seems to jump about like crazy when we make tiny
adjustments to the input parameters. If we plot how the function reacts
to combinations of the two parameters, by using them as the "X"
and
"Y" coordinates for a graph, we get a visual
description of how the pattern's boundary shifts with
position. That intricate and chaotic boundary shape gives us the
well-known shape of the standard two-dimensional Mandelbrot
set.

But
what happens if we want to be able to produce a three-dimensional
fractal with similar characteristics?
To manage that,
we'd need a similar a formula that has with three
independent parameters rather than two.

To achieve this, we can fuse the Mandelbrot
set with it's alternative sibling, the "Tricorn" fractal
(sometimes known as
the "Mandelbar" set).
These two
fractals are mathematically very similar: the key part of both
formulae has two different components, a "real
number" component that's usually plotted horizontally on
the "X"
axis, and an "imaginary number" component that's
usually plotted
vertically, on the "Y" axis, giving the
signature ( r, i ).
The only difference between the two formulae is that the "Tricorn" has
a minus sign tacked onto the front of its "imaginary"
component ( r, - i ).
Since the two functions have the same "horizontal"
formula, we can compact their four parameters into three, and plot a
three-dimensional shape (
r, i, - i ) that will generate a "Mandelbrot
set" cross-section
when the Tricorn-specific part of the formula gives zero, and a
"Tricorn set" when
the Mandelbrot contribution is zero.

We then have a three-dimensional
fractal surface enclosing a solid that can be "cut" to produce either
set.

Here's
the full three-dimensional shape:

In
this image, the shape was colour-coded according to how far
each point was from the "Mandelbrot" plane, so if we cut the solid in
half, along the middle of the central red band, we get the Mandelbrot
set.

We
might have instead chosen a different colour scheme that emphasises
the "Tricorn" plane. The version below does this, and has also
had one
quarter of the shape cut away, to make the two "standard"
cross-sections more obvious.

Here's
a couple of animations of the solid rotating in three
dimensions:

Self-similarity

Like
the Mandelbrot and Tricorn fractals, this solid shows some
self-similarity at different scales.

If
we
work leftwards along the central axis of the
Mandelbrot set,
we come across the smaller "Mandelbrot-like" shape (leftmost
image):

Cross-sections
of a smaller "island" in the combined Mandelbrot-Tricorn
solid,
taken at 0, 18, 36, 54, 66, 72 and 90 degrees

,
while the "Tricorn-like" shape on the
far right appears at exactly the same location and magnification, for
the
"Tricorn" fractal.

If
we now take our combined 3D
shape, and slice through the X-axis at different angles, this location
generate a
series of shapes that echo how that larger solid appears for
that
cross-section.
Ignoring the little offshoot strands, the smaller shape, in three
dimensions, is basically an echo of our larger solid. The sort of
self-similarity that shows up in the Mandelbrot and Tricorn
fractals appears (to some extent) in our new three-dimensional shape,
too.

Additional features

The
left-hand "bulb" of the Mandelbrot and Tricorn fractals look rather
similar, and in three dimensions we see that this part of the solid
looks suspiciously like the result of simply rotating and
distorting a shape around the X-axis.
However, when we look at the full shape, we see
other, new, features
that don't correspond to a simple morph, rotation or distortion of the
two earlier 2-D shapes.

Taking a close-up view of
the piece of the solid that's closest to us
... corresponding to the larger lower lobe of the Mandelbrot set
... we see that it appears in three dimensions to be circled by
"orbits" and
cross-connections between features and lobes that don't have direct
counterparts in either the Mandelbrot or the Tricorn
sets.

Although
most of the shape's detail appears near the Mandelbrot and
Tricorn
planes, this detail doesn't simply fade in and out . The solid includes
complex strands that circle and link features of the Mandelbrot set,
and act outside the Mandelbrot plane. Although the shape
has two fairly smooth curved faces, these regions are circled
and
overhung by "perimeter fronds" -- the blue "shoulder" in the
centre-right of the image above represents a bunch of these fronds that
are folded back on themselves to connect to the rest of the shape.

This isn't just a simple "morph" of two 2-D
fractal shapes, it's
a proper fractal solid in its own right.